System GMM: does it includes fixed effects?

[alessandro gst]

Hello everyone,
I have a panel of 97 countries over 18 years and I would like to estimate a model with country and year fixed effects.

I have estimated the model with a difference GMM in the following manner.

xtabond2 y l.y l.x l.$controls i.Year, gmm(y x $controls_endog, lag(2 5) collapse) iv(i.Year l.$controls_exogen) noleveleq small noconstant robust

And, as far as I have understood, this model addresses the issue of fixed effects.

I have also estimated the model with at SYS GMM in the following manner:

xtabond2 y l.y l.x l.$controls i.Year, gmm(y x $controls_endog, lag(2 5) collapse) iv(i.Year l.$controls_exogen) small noconstant robust

And the results appear to be very different. Among other reasons to adjudicate between these models, my main concern is whether the System GMM accounts for the country fixed effects. As reading Roodman 2009 I did not fully understand whether this is the case.

I thank you in advance for your help

Best regards.

[Sebastian Kripfganz]

The "fixed effects" are part of the error term of the untransformed level model. To account for them, you need to choose the instrumental variables such that they are uncorrelated with these unobserved time-invariant effects. For the instruments for the level model, this is usually an assumption that you need to justify.

[alessandro gst]

Dear Prof Kripfganz, thank you very much for your kind reply,
However, I am not sure I fully understand the issue. is this to say that changes in the instruments are uncorrelated with the FE?
this seems hard to believe in many cases.. it would be very useful, if you do not mind, if you could make an example that illustrates when this could be the case.
Or if there is a test to estimate whether this assumption holds.

I thank you in advance for your reply

[Sebastian Kripfganz]

It means that those instruments for the level equation must be uncorrelated with the unobserved fixed effects. For the GMM-type instruments these are typically the one-period changes of the specified variables. A sufficient condition for their validity, i.e. for them being uncorrelated with the fixed effects, is that all variables in the regression model are jointly mean stationary, which may or may not be a reasonable assumption; see Blundell and Bond (1998). All standard instruments, i.e. those provided in the iv() option, must be uncorrelated with the fixed effects as they are specified (i.e. not just their changes but the variables themselves). In your case, none of the variables in $controls_exogen is allowed to be correlated with the fixed effects. This could indeed be hard to justify in many cases.

Reference:
Blundell, R., and S. R. Bond (1998). Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87(1), 115–143.

[alessandro gst]

Thank you very much for your answer, it is much more clear now. It makes me think that many SYS GMM papers probably have some weakeneses.
I have two further doubts.
Can you confirm that even if the assumption of joint mean stationarity holds, the SYS GMM does not account for unobserved unit heterogeneity?
One more thing, is there any test you would recommend to evaluate whether the GMM instruments are joint mean stationary?

Best regards

[Sebastian Kripfganz]

The question is: What do you really mean by "account for unobserved unit heterogeneity"?

Yo do not usually test directly for mean stationarity. Instead, the Hansen overidentification test and in particular the difference-in-Hansen test on just the instruments for the level model are applied.

[alessandro gst]

I was wondering whether, if the assumption of joint mean stationarity holds, System GMM is able to account for differences between countries that are stable over time and are not covered by other regressors in the model, including (unobserved) economic, political, or cultural determinants. If I have well understood the issue, this is what fixed effects are able to control for and difference GMM is robust to this type of unobserved heterogeneity.

[Sebastian Kripfganz]

All you need for that argument is that all the instruments are valid, i.e. uncorrelated with the error term (in particular the fixed-effects component of the error term). Mean stationarity can help to ensure the validity of these results. In that sense, the system GMM estimator accounts for the fixed effects, yes.

[alessandro gst]

Thanks a lot this clearly explains a lot of issues ! also thank you for your reply on the reply on the xtdpdgmm command! I have just realized you replied there as well, for some reason I missed statalist notification on that!

[Al-Mouksit Akim]

Quick question on this issue Prof Kripfganz, what if this instrument we want to use is invariant over time? Will it be absorbed in the fixed effects as it's the case when you estimate a panel fixed regression?

[Al-Mouksit Akim]

Quick question on this issue Prof Kripfganz, what if this instrument we want to use is invariant over time? Will it be absorbed in the fixed effects as it's the case when you estimate a panel fixed regression?

[Sebastian Kripfganz]

Fixed effects (and other time-invariant variables) are not absorbed when using the GMM estimator. Instead, the strategy is to find instrumental variables which are orthogonal (or at least uncorrelated) to the fixed effects. With time-invariant instruments, you typically need a good story why those variables are uncorrelated with the fixed effects, which requires an idea about what those fixed effects (omitted time-invariant control variables) represent.